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Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization. (English) Zbl 1242.90146

Summary: Inspired by the successful applications of the stochastic optimization with second order stochastic dominance (SSD) model in portfolio optimization, we study new numerical methods for a general SSD model where the underlying functions are not necessarily linear. Specifically, we penalize the SSD constraints to the objective under Slater’s constraint qualification and then apply the well known stochastic approximation (SA) method and the level function method to solve the penalized problem. Both methods are iterative: the former requires to calculate an approximate subgradient of the objective function of the penalized problem at each iterate while the latter requires to calculate a subgradient. Under some moderate conditions, we show that w.p.1 the sequence of approximated solutions generated by the SA method converges to an optimal solution of the true problem. As for the level function method, the convergence is deterministic and in some cases we are able to estimate the number of iterations for a given precision. Both methods are applied to portfolio optimization problem where the return functions are not necessarily linear and some numerical test results are reported.

MSC:

90C15 Stochastic programming
91G10 Portfolio theory
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[1] Baker, B.M.; Sheasby, J., Accelerating the convergence of subgradient optimization, European journal of operational research, 117, 136-144, (1999) · Zbl 0998.90056
[2] Branda, M., Reformulation of general chance constrained problems using the penalty functions, vol. 2, (2010), Stochastic Programming E-print Series
[3] Clarke, F.H., Optimization and non-smooth analysis, (1983), Wiley New York
[4] Dentcheva, D.; Ruszczyński, A., Optimization with stochastic dominance constraints, SIAM journal of optimization, 13, 548-566, (2003) · Zbl 1055.90055
[5] Dentcheva, D.; Ruszczyński, A., Portfolio optimization with stochastic dominance constraints, Journal of banking and finance, 30, 433-451, (2007)
[6] Dentcheva, D.; Ruszczyński, A., Composite semi-infinite optimization, Control and cybernetics, 36, 634-646, (2007) · Zbl 1166.49025
[7] Dentcheva, D.; Ruszczyński, A., Optimization with multivariate stochastic dominance constraints, Mathematical programming series B, 117, 111-127, (2009) · Zbl 1221.90069
[8] Dupačová, J.; Gaivoronski, A.; Kos, Z.; Szantai, T., Stochastic programming in water management: A case study and a comparison of solution techniques, European journal of operational research, 52, 28-44, (1991) · Zbl 0726.90048
[9] Ermoliev, Y., Random optimization and stochastic programming, Lecture notes in mathematics, 112, 104-115, (1970)
[10] Ermoliev, Y., Stochastic programming methods, (1976), Nauka Moscow
[11] Ermoliev, Y., Stochastic quasi-gradient methods and their application to system optimization, Stochastics, 9, 1-36, (1983) · Zbl 0512.90079
[12] Ermoliev, Y.; Wets, R.J.B., Numerical techniques for stochastic optimization, (1988), Springer-Verlag Berlin
[13] Ermoliev, Y.; Norkin, V.I., On nonsmooth and discontinuous problems of stochastic systems optimization, European journal of operational research, 101, 230-244, (1997) · Zbl 0929.90060
[14] Fábián, C.I.; Mitra, G.; Roman, D., Processing second order stochastic dominance models using cutting-plane representations, Mathematical programming series A, 1-25, (2009)
[15] Fishburn, P.C., Decision and value theory, (1964), John Wiley & Sons New York · Zbl 0149.16203
[16] Gustafson, S., Investigating semi-infinite programs using penalty function and Lagrangian methods, Australian mathematical society, series B, 28, 158-169, (1986) · Zbl 0616.90062
[17] Larsson, T.; Patriksson, M.; Strömberg, A., Conditional subgradient optimization – theory and application, European journal of operational research, 88, 382-403, (1996) · Zbl 0913.90225
[18] Lemarechal, C.; Nemirovskii, A.; Nestrov, Y., New variants of bundle methods, Mathematical programming, 69, 111-147, (1995) · Zbl 0857.90102
[19] Homem-de-Mello, T.; Mehrota, S., A cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints, SIAM journal of optimization, 20, 1250-1273, (2009) · Zbl 1198.90291
[20] Y. Lin, H. Xu, Stability and sensitivity analysis of stochastic programs with second order dominance constraints, Stochastic Programming E-Print Series. <http://edoc.hu-berlin.de/docviews/abstract.php?id=36988>, 2010.
[21] Ogryczak, W.; Ruszczyński, A., From stochastic dominance to Mean-risk models: semideviations as risk measures, European journal of operational research, 116, 33-50, (1999) · Zbl 1007.91513
[22] Poljak, B.T., Nonlinear programming methods in the presence of noise, Mathematical programming, 14, 87-97, (1978) · Zbl 0379.90080
[23] Robbins, H.; Monro, S., A stochastic approximation method, Annual mathematical statistics, 22, 400-407, (1951) · Zbl 0054.05901
[24] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton, NJ · Zbl 0229.90020
[25] Roman, D.; Darby-Dowman, K.; Mitra, G., Portfolio construction based on stochastic dominance and target return distributions, Mathematical programming, 108, 541-569, (2000) · Zbl 1138.91476
[26] Ruszczyński, A.; Shapiro, A., Stochastic programming, Handbook in operations research and management science, (2003), Elsevier
[27] A. Shapiro, Monte Carlo sampling methods, in: Stochastic Programming, Handbook in Operations Research and Management Science, Stochastic Programming, vol. 10, 2003, pp. 353-425.
[28] Whitmore, G.A.; Findlay, M.C., Stochastic dominance: an approach to decison-making under risk, (1978), D.C. Health Lexington, MA
[29] Xu, H., Level function method for quasiconvex programming, Journal of optimization theory and applications, 108, 407-437, (2001) · Zbl 1041.90038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.